It can be commented that the OP's form isn't good...but begs the question...how do you develop a better pedal stroke where you can turn higher RPM without bouncing. I believe greg offered good advice. I think about this when I ride in fact. I try to work on my pedal stroke and it has paid dividends in speed and endurance I believe.

Just pushing down isn't a good formula for a pedal stroke OP.
A couple of visuals are:
-Like scraping the mud off the bottom of your shoes at 6 o'clock.
- Pedal in circles.

If you think about it, pushing down hard on right side, can be inhibited by too much weight on the backside of left foot between 6 and 12 o'clock which opposes pushing down on the opposite side because the crank arms are connected of course. If you think about spinning both feet tangential to the chainring, then there is less chop between power zone and dead zone and hence less bounce.
This for me at least creates more natural power with less tension or force in my legs. More effortless speed than focus on just pushing down harder. Focus on pushing down without thought to pedaling in circles inhibits the upstroke which subtracts power from the down stroke in the power zone on the opposite side.